A class of dissimilarity semimetrics for preference relations

In practical decision-making, it seems clear that if we hope to make an optimal or at least defensible decision, we must weigh our alternatives against each other and come to a principled judgment between them. In the formal literature of classical decision theory, it is taken as an indispensable axiom that cardinal rankings of alternatives be defined for all possible alternatives over which we might have to decide. Whether there are any items " beyond compare " is thus a crucial question for decision theorists to consider when constructing a formal framework. At the very least, it seems problematic to presuppose that no such incommensurability is possible on the grounds that it would make formalizing axioms for decision-making more difficult, or even intractable. With this in mind, I plan to argue in this paper that a formal notion of comparability can be introduced to the classical understanding of preference relations such that the question of comparability between alternatives can be taken non-trivially. Building on the work of Richard Bradley and Ruth Chang, I argue that the comparability relation should be understood to be transitive but not complete. I contend that this understanding of comparability within decision theory can explain both why we believe that some alternatives may be incommensurable, yet we are still able to make justified decisions despite incomplete preference relations. In Section I, I lay the groundwork for understanding the conceptual relationship between comparability and commensurability with respect to decision-making. In Section II, I will argue that Bradley's definition of the preference relation with comparability leads to absurdity and contradiction due to a small oversight, which I propose to remedy. Then,

Decision Analysis

This paper characterizes lexicographic preferences over alternatives that are identified by a finite number of attributes. Our characterization is based on two key concepts: a weaker notion of continuity called “mild continuity” (strict preference order between any two alternatives that are different with respect to every attribute is preserved around their small neighborhoods) and an “unhappy set” (any alternative outside such a set is preferred to all alternatives inside). Three key aspects of our characterization are as follows: (i) we use continuity arguments; (ii) we use the stepwise approach of looking at two attributes at a time; and (iii) in contrast with the previous literature, we do not impose noncompensation on the preference and consider an alternative weaker condition.

Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles

Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure of set of preferences by "replacing" the elements ι ∈ I with their inverse images p −1 (ι) considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences A whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.

Journal of Interdisciplinary Mathematics, 2004

In decision theory several preference structures are used for modeling coherence and rational behavior. In this paper we establish, from an algebraic approach, characterizations of some general properties involving preference and indifference relations, as well as the more common preference structures used in the literature.

Notre Dame Journal of Formal Logic, 1985

Some years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. 2 Definitions Throughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by: x <* y <-> x <* y & ~y <* x X =* y +-> x <* y & y <* χ m Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be *Research in part supported by USA National Science Foundation Grant MCS 8003254.

Rivista Di Matematica Per Le Scienze Economiche E Sociali, 2002

Journal of Mathematical Psychology, 2014

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Mathematical Modelling, 1987

Theories that attempt to represent decision makers' preferences are usually based on the axiom of transitivity.

Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), 2019

Convexity of preferences is a canonical assumption in economic theory. In this paper we consider a generalized definition of convex preferences that relies on the abstract notion of convex space.

Mathematical Social Sciences, 1995

We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).